# First tutorial on the Projector Augmented-Wave (PAW) technique¶

## Projector Augmented-Wave technique, how to use it?¶

This tutorial aims at showing how to perform a calculation within the Projector Augmented-Wave (PAW) method.

You will learn how to launch a PAW calculation and what are the main input variables that govern convergence and numerical efficiency. You are supposed to know how to use ABINIT with Norm-Conserving PseudoPotentials (NCPP).

This tutorial should take about 1.5 hour.

Note

Supposing you made your own install of ABINIT, the input files to run the examples are in the ~abinit/tests/ directory where ~abinit is the absolute path of the abinit top-level directory. If you have NOT made your own install, ask your system administrator where to find the package, especially the executable and test files.

To execute the tutorials, create a working directory (Work*) and copy there the input files of the lesson.

Most of the tutorials do not rely on parallelism (except specific tutorials on parallelism). However you can run most of the tutorial examples in parallel, see the topic on parallelism.

In case you work on your own PC or workstation, to make things easier, we suggest you define some handy environment variables by executing the following lines in the terminal:

export ABI_HOME=Replace_with_the_absolute_path_to_the_abinit_top_level_dir
export PATH=$ABI_HOME/src/98_main/:$PATH
export ABI_TESTS=$ABI_HOME/tests/ export ABI_PSPDIR=$ABI_TESTS/Psps_for_tests/  # Pseudopotentials used in examples.


Examples in this tutorial use these shell variables: copy and paste the code snippets into the terminal (remember to set ABI_HOME first!). The ‘export PATH’ line adds the directory containing the executables to your PATH so that you can invoke the code by simply typing abinit in the terminal instead of providing the absolute path.

## 1. Summary of the PAW method¶

The Projector Augmented-Wave approach has been introduced by Peter Blochl in 1994:

The Projector Augmented-Wave method is an extension of augmented wave methods and the pseudopotential approach, which combines their traditions into a unified electronic structure method”. It is based on a linear and invertible transformation (the PAW transformation) that connects the “true” wavefunctions $\Psi$ with “auxiliary” (or “pseudo”) soft wavefunctions $\tPsi$:

$$\label{eq:PAW_true_wavefunction} |\Psi\rangle = |\tPsi\rangle + \sum_i \sum_a \Bigl(|\phi_i^a\rangle - |\tphi_i^a\rangle \Bigr) \,\langle \tprj_i^a|\tPsi\rangle = |\tPsi\rangle + \sum_a |\Psi^1_a\rangle - |\tPsi_a^1\rangle$$

This relation is based on the definition of augmentation regions (atomic spheres of radius $r_c$), around the atoms in which the partial waves $|\phi_i\rangle$ form a basis of atomic wavefunctions; $|\tphi_i\rangle$ are pseudized partial waves (obtained from $|\phi_i\rangle$), and $|\tprj_i\rangle$ are dual functions of the $|\tphi_i\rangle$ called projectors. It is therefore possible to write every quantity depending on $\Psi_n$ (density, energy, Hamiltonian) as a function of $\tPsi_n$ and to find $\tPsi_n$ by solving self-consistent equations.

The PAW method has two main advantages:

1. From $\tPsi$, it is always possible to obtain the true all electron wavefunction $\Psi$,
2. The convergence is comparable to an UltraSoft PseudoPotential (USPP) one.

From a practical point of view, a PAW calculation is rather similar to a Norm-Conserving PseudoPotential one. Most noticeably, one will have to use a special atomic data file (PAW dataset) that contains the $\phi_i$, $\tphi_i$ and $\tprj_i$ and that plays the same role as a pseudopotential file.

Tip

It is highly recommended to read the following papers to understand correctly the basic concepts of the PAW method: [Bloechl1994] and [Kresse1999]. The implementation of the PAW method in ABINIT is detailed in [Torrent2008], describing specific notations and formulations

## 2. Using PAW with ABINIT¶

Before continuing, you might consider to work in a different subdirectory as for the other tutorials. Why not Work_paw1? In what follows, the name of files are mentioned as if you were in this subdirectory. All the input files can be found in the $ABI_TESTS/tutorial/Input directory. Important You can compare your results with reference output files located in $ABI_TESTS/tutorial/Refs and $ABI_TESTS/tutorial/Refs/tpaw1_addons directories (for the present tutorial they are named tpaw1_*.abo). The input file tpaw1_1.in is an example of a file to be used to compute the total energy of diamond at the experimental volume (within the LDA exchange-correlation functional). You might get the corresponding output file (it is available in$ABI_TESTS/tutorial/Refs/tpaw1_1.abo). Copy the files *tpaw1_1.in in your work directory,

Important

You may have to change the path to reach the Psps_for_tests repository. For this, modify the varaible pp_dirpath in the input file.

and execute:

abinit  tpaw1_1.in > log 2> err &


The code should run very quickly. In the meantime, you can read the input file and see that there is no PAW input variable.

Now, open the tpaw1_1.in file and change the line iwth pseudos information: replace the PseudosTM_pwteter/6c.pspnc file with C.LDA_PW-JTH.xml. Run the code again:

abinit tpaw1_1.in > log 2> err &


Your run should stop almost immediately! The input file, indeed, is missing the mandatory argument pawecutdg!!

pawecutdg 50


to tpaw1_1.in and run it again. Now the code completes successfully.

Note

The time needed for the PAW run is greater than the time needed for the Norm-Conserving PseudoPotential run; indeed, at constant value of plane wave cut-off energy ecut PAW requires more computational resources:

• the on-site contributions have to be computed,
• the nonlocal contribution of the PAW dataset uses 2 projectors per angular momentum, while the nonlocal contribution of the Present Norm-Conserving Pseudopotential uses only one.

However, as the plane wave cut-off energy required by PAW is much smaller than the cut-off needed for the Norm-Conserving PseudoPotential (see next section), a PAW calculation will actually require less CPU time.

Let’s open the output file (tpaw1_1.abo) and have a look inside. Compared to an output file for a Norm-Conserving PseudoPotential run, an output file for PAW contains the following specific topics:

• At the beginning of the file, some specific default PAW input variables (ngfftdg, pawecutdg, and useylm), mentioned in the section:
    -outvars: echo values of preprocessed input variables --------

• The use of two FFT grids, mentioned in:
Coarse grid specifications (used for wave-functions):

getcut: wavevector=  0.0000  0.0000  0.0000  ngfft=  18  18  18
ecut(hartree)=     15.000   => boxcut(ratio)=   2.17276

Fine grid specifications (used for densities):

getcut: wavevector=  0.0000  0.0000  0.0000  ngfft=  32  32  32
ecut(hartree)=     50.000   => boxcut(ratio)=   2.10918

• A specific description of the PAW dataset (you might follow the tutorial PAW2, devoted to the building of the PAW atomic data, for a complete understanding of the file):
Pseudopotential format is: paw10
basis_size (lnmax)=  4 (lmn_size=  8), orbitals=   0   0   1   1
- mesh 1: r(i)=AA*[exp(BB*(i-1))-1], size= 500 , AA= 0.33742E-02 BB= 0.20146E-01
Shapefunction is SIN type: shapef(r)=[sin(pi*r/rshp)/(pi*r/rshp)]**2
Radius for shape functions =  1.28249356
mmax=  500
Radial grid used for partial waves is grid 1
Radial grid used for projectors is grid 1
Radial grid used for (t)core density is grid 1
Radial grid used for Vloc is grid 1
Radial grid used for LDA-1/2 potential is grid 1
Radial grid used for pseudo valence density is grid 1

• After the SCF cycle section: The value of the integrated compensation charge evaluated by two different numerical methodologies; 1- computed in the augmentation regions on the “spherical” grid, 2- computed in the whole simulation cell on the “FFT” grid… A discussion on these two values will be done in a forthcoming section.
PAW TEST:
==== Compensation charge inside spheres ============
The following values must be close to each other ...
Compensation charge over spherical meshes =      0.263855884339479
Compensation charge over fine fft grid    =      0.263852893893634

• Information concerning the non-local term (pseudopotential strength $D_{ij}$) and the spherical density matrix (augmentation wave occupancies $\rho_{ij}$):
==== Results concerning PAW augmentation regions ====

Total pseudopotential strength Dij (hartree):
Atom #  1
...
Atom #  2
...

Augmentation waves occupancies Rhoij:
Atom #  1
...
Atom #  2
...

• At the end of the file we find the decomposition of the total energy both by direct calculation and double counting calculation:
--- !EnergyTerms
iteration_state     : {dtset: 1, }
comment             : Components of total free energy in Hartree
kinetic             :  6.90446929441886E+00
hartree             :  9.63708091533161E-01
xc                  : -4.29580646918379E+00
Ewald energy        : -1.27864121210521E+01
psp_core            :  9.19865486989434E-01
local_psp           : -4.67413594752577E+00
spherical_terms     :  1.44220243759604E+00
total_energy        : -1.15261092272241E+01
total_energy_eV     : -3.13641382594284E+02
...

--- !EnergyTermsDC
iteration_state     : {dtset: 1, }
comment             : '"Double-counting" decomposition of free energy'
band_energy         :  3.07962181105384E-01
Ewald energy        : -1.27864121210521E+01
psp_core            :  9.19865486989434E-01
xc_dc               : -7.39239833326780E-01
spherical_terms     :  7.71714533951319E-01
total_energy_dc     : -1.15261097523327E+01
total_energy_dc_eV  : -3.13641396883215E+02
...


Note

The PAW total energy is not the equal to the one obtained in the Norm-Conserving PseudoPotential case: in the Norm-Conserving PseudoPotential case, the energy reference has been arbitrarily modified by the pseudopotential construction procedure. Comparing total energies computed with different PAW potentials is more meaningful: most of the parts of the energy are calculated exactly, and in general you should be able to compare numbers for (valence) energies between different PAW potentials or different codes.

## 3. Convergence with respect to the plane-wave basis cut-off¶

As in the usual Norm-Conserving PseudoPotential case, the critical convergence parameter is the cut-off energy defining the size of the plane-wave basis.

### 3.a. Convergence with respect to ecut in the Norm-Conserving PseudoPotential case¶

The input file tpaw1_2.in contains data to be used to compute the convergence in ecut for diamond (at experimental volume). There are 9 datasets, with increasing ecut values from 8 Ha to 24 Ha. You might use the tpaw1_2.in file (with a standard Norm-Conserving PseudoPotential), and run:

abinit tpaw1_2.in > log 2> err &


You should obtain the following total energy values (see tpaw1_2.abo):

etotal1  -1.1628880677E+01
etotal2  -1.1828052470E+01
etotal3  -1.1921833945E+01
etotal4  -1.1976374633E+01
etotal5  -1.2017601960E+01
etotal6  -1.2046855404E+01
etotal7  -1.2062173253E+01
etotal8  -1.2069642342E+01
etotal9  -1.2073328672E+01


You can check that the etotal convergence (at the 1 mHartree level) is not achieved for ecut = 24 Hartree.

### 3.b. Convergence with respect to ecut in the PAW case¶

Use the same input files as in section 1.a. Again, modify the last line of tpaw1_2.in, replacing the PseudosTM_pwteter/6c.pspnc file by C.LDA_PW-JTH.xml. Run the code again and open the output file. You should obtain the values:

etotal1    -1.1404413200E+01
etotal2    -1.1496546303E+01
etotal3    -1.1518699851E+01
etotal4    -1.1524923431E+01
etotal5    -1.1526676260E+01
etotal6    -1.1526950267E+01
etotal7    -1.1526965855E+01
etotal8    -1.1527043191E+01
etotal9    -1.1527176114E+01


You can check that:

The etotal convergence (at 1 mHartree) is achieved for 14 <= ecut <= 16 Hartree (etotal5 is within 1 mHartree of the final value);

With the same input parameters, for diamond, a PAW calculation needs a lower cutoff, compared to a calculation with NCPPs.

## 4. Convergence with respect to the double grid FFT cut-off¶

In a NCPP calculation, the plane wave density grid should be (at least) twice bigger than the wavefunctions grid, in each direction. In a PAW calculation, the plane wave density grid is tunable thanks to the input variable pawecutdg (PAW: ECUT for Double Grid). This is mainly needed to allow the mapping of densities and potentials, located in the augmentation regions (spheres), onto the global FFT grid. The number of points of the Fourier grid located in the spheres must be large enough to preserve a minimal accuracy. It is determined from the cut-off energy pawecutdg. An alternative is to use directly the input variable ngfftdg. One of the most sensitive objects affected by this “grid transfer” is the compensation charge density; its integral over the augmentation regions (on spherical grids) must cancel with its integral over the whole simulation cell (on the FFT grid).

Use now the input file tpaw1_3.in . The only difference with the tpaw1_2.in file is that ecut is fixed to 12 Ha, while pawecutdg runs from 12 to 39 Ha.

Launch ABINIT with these files; you should obtain the values (file tpaw1_3.abo):

etotal1    -1.1518150865E+01
etotal2    -1.1518273022E+01
etotal3    -1.1518608846E+01
etotal4    -1.1518744552E+01
etotal5    -1.1518729256E+01
etotal6    -1.1518683931E+01
etotal7    -1.1518660738E+01
etotal8    -1.1518667056E+01
etotal9    -1.1518682154E+01
etotal10   -1.1518697442E+01


We see that the variation of the energy wit respect to the pawecutdg parameter is well below the 1 mHa level. In principle, it should be sufficient to choose pawecutdg = 12 Ha in order to obtain an energy change lower than 1 mHa. In practice, it is better to keep a security margin. Here, for pawecutdg = 24 Ha (5th dataset), the energy change is lower than 0.001 mHa: this choice will be more than enough.

Note

Note the steps in the convergence. They are due to sudden changes in the grid size (see the output values for ngfftdg) which do not occur for each increase of pawecutdg. To avoid troubles due to these steps, it is better to choose a value of pawecutdg slightly higher.

The convergence of the compensation charge has a similar behaviour; it is possible to check it in the output file, just after the SCF cycle by looking at:

PAW TEST:
==== Compensation charge inside spheres ============
The following values must be close to each other ...
Compensation charge over spherical meshes =      0.252496383260266
Compensation charge over fine fft grid    =      0.252495562362116


The two values of the integrated compensation charge density must be close to each other. Note that, for numerical reasons, they cannot be exactly the same (integration over a radial grid does not use the same scheme as integration over a FFT grid).

Additional test: We want now to check the convergence with respect to ecut with a fixed value pawecutdg = 24 Ha. Let’s modify tpaw1_2.in file, setting pawecutdg to 24 Ha, and let’s launch ABINIT again. You should obtain the values:

etotal1    -1.1404443407E+01
etotal2    -1.1496575772E+01
etotal3    -1.1518729256E+01
etotal4    -1.1524953006E+01
etotal5    -1.1526706024E+01
etotal6    -1.1526980122E+01
etotal7    -1.1526995701E+01
etotal8    -1.1527072969E+01
etotal9    -1.1527205784E+01


You can check again that:

The etotal convergence (at the 1 mHartree level) is achieved for 14 <= ecut <= 16 Hartree;

Note

Associated with the input variable pawecutdg is the input variable ngfftdg: it defines the size of the FFT grid associated with pawecutdg. Note that pawecutdg is only useful to define the FFT grid for the density in a convenient way. You can therefore tune directly ngfftdg to define the size of the FFT grid for the density.

Note

Although pawecutdg should always be checked, in practice, a common use it to put it bigger than ecut and keep it constant during all calculations. Increasing pawecutdg slightly changes the CPU execution time, but above all it is memory-consuming. Note that, if ecut is already high, there is no need for a high pawecutdg.

Important

When testing ecut convergency, pawecutdg has to remain constant to obtain consistent results.

## 5. Plotting PAW contributions to the Density of States (DOS)¶

We now use the input file tpaw1_4.in file. ABINIT is used to compute the Density Of State (DOS) (see the prtdos keyword in the input file). Also note that more k-points are used in order to increase the accuracy of the DOS. ecut is set to 12 Ha, while pawecutdg is 24 Ha.

Launch the code with these files; you should obtain the tpaw1_4.abo and the DOS file (tpaw1_4o_DOS):

abinit tpaw1_4.in > log 2> err &


You can plot the DOS file if you want; for this purpose, use a graphical tool and plot column 3 with respect to column 2. If you use the xmgrace tool, launch:

xmgrace -block tpaw1_4o_DOS -bxy 1:2


At this stage, you have a usual Density of State plot; nothing specific to PAW.

Now, edit the tpaw1_4.in file, comment the “prtdos 1” line, and uncomment (or add):

prtdos 3  pawprtdos 1 natsph 1 iatsph 1 ratsph 1.51


prtdos 3 now requires the output of the projected DOS; natsph 1 iatsph 1 ratsph 1.51 selects the first carbon atom as the center of projection, and sets the radius of the projection area to 1.51 atomic units (this is exactly the radius of the PAW augmentation regions: generally the best choice). The pawprtdos 1 is specific to PAW. With this option, ABINIT should compute all the contributions to the projected DOS.

Let us remember that:

\begin{equation*} |\Psi\rangle = |\tPsi\rangle + \sum_i \sum_a \Bigl(|\phi_i^a\rangle - |\tphi_i^a\rangle \Bigr) \,\langle \tprj_i^a|\tPsi\rangle \end{equation*}

Within PAW, the total projected DOS has 3 contributions:

1. The smooth plane-waves (PW) contribution (from $|\tPsi\rangle$),
2. The all-electron on-site (AE) contribution (from $\langle \tprj^a_i|\tPsi\rangle |\phi_i^a\rangle$),
3. The pseudo on-site (PS) contribution (from $\langle \tprj^a_i|\tPsi\rangle |\tphi_i^a\rangle$).

Launch ABINIT again (with the modified input file). You get a new DOS file, named tpaw1_4o_DOS_AT0001. You can edit it and look inside; it contains the 3 PAW contributions (mentioned above) for each angular momentum. In the diamond case, only $l = 0$ and $l = 1$ momenta are to be considered.

Now, plot the file, using the 7th, 12th and 17th columns with respect to the 2nd one; it plots the 3 PAW contributions for $l = 0$ (the total DOS is the sum of the three contributions). If you use the xmgrace tool, launch:

xmgrace -block tpaw1_4o_DOS_AT0001 -bxy 1:7 -bxy 1:12 -bxy 1:17


You should get this:

As you can see, the smooth PW contribution and the PS on-site contribution are close. At basis completeness, they should cancel; we could approximate the DOS by the AE on-site part taken alone. That’s exactly the purpose of the pawprtdos = 2 option; in that case, only the AE on-site contribution is computed and given as a good approximation of the total projected DOS. The main advantage of this option is that the computing time is greatly reduced (the DOS is instantaneously computed).

However, as you will see in the next section, this approximation is only valid when:

1. The $\tphi_i$ basis is complete enough
2. The electronic density is mainly contained in the sphere defined by ratsph.

## 6. Testing the completeness of the PAW partial wave basis¶

In the previous section we used a “standard” PAW dataset, with 2 partial waves per angular momentum. It is generally the best compromise between the completeness of the partial wave basis and the efficiency of the PAW dataset (the more partial waves you have, the longer the CPU time used by ABINIT is).

Let’s have a look at the $ABI_PSPDIR/C.LDA_PW-JTH.xml file. The tag “<valence_states” has 4 “<state” lines. This indicates the number of partial waves (4) and their $l$ angular momentum. In the present file, there are two l = 0 partial waves and two l = 1 partial waves. Now, let’s open the$ABI_PSPDIR/C.LDA_PW-2proj-JTH.xml and \$ABI_PSPDIR/C.LDA_PW-6proj-JTH.xml files. In the first file, only one partial wave per $l$ is present; in the second one, 3 partial waves per $l$ are present. The completeness of the partial wave basis increases when you use C.LDA_PW-2proj-JTH.xml, C.LDA_PW-JTH.xml and C.LDA_PW-6proj-JTH.xml.

Now, let us plot the DOS for the two new PAW datasets.

1. Save the existing tpaw1_4o_DOS_AT0001 file, naming it f.i. tpaw1_4o_4proj_DOS_AT0001.
2. Open the tpaw1_4.in file and modify it in order to use the C.LDA_PW-2proj-JTH.xml PAW dataset.
3. Launch ABINIT.
4. Save the new tpaw1_4o_DOS_AT0001 file, naming it f.i. tpaw1_4o_2proj_DOS_AT0001.
5. Open the tpaw1_4.in file and modify it in order to use the C.LDA_PW-6proj-JTH.xml PAW dataset.
6. Launch ABINIT again.
7. Save the new tpaw1_4o_DOS_AT0001 file, naming it f.i. tpaw1_4o_6proj_DOS_AT0001.

Then, plot the contributions to the projected DOS for the two new DOS files. You should get:

Adding the DOS obtained in the previous section to the comparison, you immediately see that the superposition of the plane wave part DOS (PW) and the PS on-site DOS depends on the completeness of the partial wave basis!

Now, you can have a look at the 3 output files (one for each PAW dataset) for instance in a comparison tool. A way to estimate the completeness of the partial wave basis is to compare derivatives of total energy; if you look at the stress tensor:

For the 2 partial-wave basis:   2.88568213E-04  2.88568213E-04 2.88568213E-04  0.  0.  0.
For the 4 partial-wave basis:   4.97807406E-04  4.97807406E-04 4.97807406E-04  0.  0.  0.
For the 6 partial-wave basis:   5.38983766E-04  5.38983766E-04 5.38983766E-04  0.  0.  0.


The 2 partial-wave basis is clearly not complete; the 4 partial-wave basis results are correct. Such a test is useful to estimate the precision we can expect on the stress tensor (at least due to the partial wave basis completeness).

You can compare other results in the 3 output files: total energy, eigenvalues, occupations…

Note: if you want to learn how to generate PAW datasets with different partial wave basis, you might follow the tutorial on generating PAW datasets(PAW2).

## 7. Checking the validity of PAW results¶

The validity of our computation has to be checked by comparison, on known structures, with known results. In the case of diamond, lots of computations and experimental results exist. Very important remark: the validity of PAW calculations (completeness of plane wave basis and partial wave basis) should always be checked by comparison with all-electrons computations or with other existing PAW results; it should not be done by comparison with experimental results. As the PAW method has the same accuracy than all-electron methods, results should be very close.

In the case of diamond, all-electron results can be found f.i. in [Holzwarth1997]. All-electron equilibrium parameters for diamond (within Local Density Approximation) obtained with the FP-LAPW WIEN2K code are:

a0 = 3.54 angstrom
B = 470 GPa


Experiments give:

a0 = 3.56 angstrom
B = 443 GPa


Let’s test with ABINIT. We use now the input file tpaw1_5.in file and we run ABINIT to compute values of _etotal- for several cell parameters around 3.54 angstrom, using the standard PAW dataset.

abinit tpaw1_5.in > log 2> err &


From the tpaw1_5.abo file, you can extract the 7 values of acell and 7 values of etotal, then put them into a file and plot it with a graphical tool. You should get:

From this curve, you can extract the cell values of $a_0$ and $B$ (with the method of your choice, for example by a Birch-Murnhagan spline fit). You get:

a0 = 3.53 angstrom B = 469.5 GPa


These results are in excellent agreement with FP-LAPW ones!

### 8.a. Overlap of PAW augmentation regions¶

In principle, the PAW formalism is only valid for non-overlapping augmentation spherical regions. But, in usual cases, a small overlap between spheres is acceptable. By default, ABINIT checks that the distances between atoms are large enough to avoid overlap; a “small” voluminal overlap of 5% is accepted by default. This value can be tuned with the pawovlp input keyword. The overlap check can even be by-passed with pawovlp=-1 (not recommended!).

Warning

While a small overlap can be acceptable for the augmentation regions, an overlap of the compensation charge densities has to be avoided. The compensation charge density is defined by a radius (named $r_{shape}$ in the PAW dataset) and an analytical shape function. The overlap related to the compensation charge radius is checked by ABINIT and a WARNING is eventually printed.

Also note that you can control the compensation charge radius and shape function while generating the PAW dataset (see tutorial on generating PAW datasets(PAW2)).

### 8.b. Mixing scheme for the Self-Consistent cycle; decomposition of the total energy¶

The use of an efficient mixing scheme in the self-consistent loop is a crucial point to minimize the number of steps to achieve convergence. This mixing can be done on the potential or on the density. By default, in a Norm-Conserving PseudoPotential calculation, the mixing is done on the potential; but, for technical reasons, this choice is not optimal for PAW calculations. Thus, by default, the mixing is done on the density when PAW is activated. The mixing scheme can be controlled by the iscf variable (see the different options of this input variable). To compare both schemes, you can edit the tpaw1_1.in file and try iscf = 7 or 17 and compare the behaviour of the SC cycle in both cases; as you can see, the final total energy is the same but the way to reach it is completely different.

Now, have a look at the end of the file and focus on the Components of totalfree energy; the total energy is decomposed according to two different schemes (direct and double counting); at very high convergence of the SCF cycle the potential/density residual is very small and these two values should be the same. But it has been observed that the converged value was reached more rapidly by the direct energy, when the mixing is on the potential, and by the double counting energy when the mixing is on the density. Thus, by default, in the output file is to print the direct energy when the mixing is on the potential, and the double counting energy when the mixing is on the density.

Also note that PAW partial waves occupancies $\rho_{ij}$ also are mixed during the SC cycle; by default, the mixing is done in the same way as the density.

### 8.c. PAW+U for correlated materials¶

If the system under study contains strongly correlated electrons, the DFT+U method can be useful. It is controlled by the usepawu, lpawu, upawu and jpawu input keywords. Note that the formalism implemented in ABINIT is approximate, i.e. it is only valid if:

1. The $\tphi^a_i$ basis is complete enough;
2. The electronic density is mainly contained in the PAW sphere.

The approximation done here is the same as the one explained in the 5th section of this tutorial: considering that smooth PW contributions and PS on-site contributions are closely related, only the AE on-site contribution is computed; it is indeed a very good approximation.

Converging a Self-Consistent Cycle, or ensuring the global minimum is reached, with PAW+U is sometimes difficult. Using usedmatpu and dmatpawu can help.

### 8.d. Printing volume for PAW¶

If you want to get more detailed output concerning the PAW computation, you can use the pawprtvol input keyword. It is particularly useful to print details about pseudopotential strength $D_{ij}$ or partial waves occupancies $\rho_{ij}$.

### 8.e. Additional PAW input variables¶

Looking at the PAW variable set, you can find the description of additional input keywords related to PAW. They are to be used when tuning the computation, in order to gain accuracy or save CPU time.

Warning

In a standard computation, these variables should not be modified!

Variables that can be used to gain accuracy (in ascending order of importance)

pawxcdev
Control of the accuracy of exchange-correlation on-site potentials (try pawxcdev = 2 to increase accuracy).
mqgriddg
Control of the accuracy of spline fits to transfer densities/potentials from FFT grid to spherical grid.
pawnzlm
Control how the symmetries are applied to compute the moments of spherical densities.

Variables that can be used to save memory (in ascending order of importance)

pawstgylm
Control of the storage of spherical harmonics around atoms.
pawmixdg
Control the grid used to mix the potential/density during SCF cycles.
pawlcutd
Control of the number of angular momenta taken into account in on-site densities.
pawlmix
Control of the number of $\rho_{ij}$ components to be mixed during SCF cycle.

Variables that can be used to save CPU time (in ascending order of importance)

pawnhatxc
Control of the numerical treatment of gradients in case of GGA.
pawstgylm
Control of the storage of spherical harmonics around atoms.
pawlcutd
Control of the number of angular momenta taken into account in on-site densities.
pawlmix
Control of the number of $\rho_{ij}$ components to be mixed during SCF cycle.
bxctmindg
Can be used to decrease the size of fine FFT grid for a given value of pawecutdg.

Note

The above list is not exhaustive. Several other keywords can be used to tune ABINIT PAW calculations.